443 research outputs found
Existence of weak solutions for the generalized Navier-Stokes equations with damping
In this work we consider the generalized Navier-Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any and any sigma > 1, where q is the exponent of the diffusion term and sigma is the exponent which characterizes the damping term.MCTES, Portugal [SFRH/BSAB/1058/2010]; FCT, Portugal [PTDC/MAT/110613/2010]info:eu-repo/semantics/publishedVersio
Optimal well-posedness for the inhomogeneous incompressible Navier-Stokes system with general viscosity
In this paper we obtain new well-possedness results concerning a linear
inhomogenous Stokes-like system. These results are used to establish local
well-posedness in the critical spaces for initial density and
velocity such that
,
, , for the inhomogeneous incompressible Navier-Stokes
system with variable viscosity. To the best of our knowledge, regarding the
case, this is the first result in a truly critical framework for which one
does not assume any smallness condition on the density
On a nonlocal degenerate parabolic problem
Conditions for the existence and uniqueness of weak solutions for a class of
nonlinear nonlocal degenerate parabolic equations are established. The
asymptotic behaviour of the solutions as time tends to infinity are also
studied. In particular, the finite time extinction and polynomial decay
properties are proved
Anisotropic parabolic equations with variable nonlinearity
We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions
Existence and large time behavior for generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluids
Generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible
fluids are considered in this work. We assume that, in the momentum equation, the diffusion
and relaxation terms are described by two distinct power-laws. Moreover, we assume that the
momentum equation is perturbed by an extra term, which, depending on whether its signal is
positive or negative, may account for the presence of a source or a sink within the system. For
the associated initial-boundary value problem, we study the existence of weak solutions as well
as the large time behavior of the solutions.Portuguese Foundation for Science and Technology: UID/MAT/04561/2019info:eu-repo/semantics/publishedVersio
A class of electromagnetic p-curl systems: blow-up and finite time extinction
We study a class of -curl systems arising in electromagnetism, for , with nonlinear source or sink terms. Denoting by the magnetic field, the source terms considered are of the form , with . Existence of local or global solutions is proved depending on values of and . The blow-up of local solutions is also studied.
The sink term is of the form , with . Existence and finite time extinction of solutions are proved, for certain values of and .The first author was supported partially by the research project PTDC/MAT/110613/2009, FCT, Portugal. The research of the second and third authors was partially supported by CMAT-"Centro de Matemetica da Universidade do Minho", financed by FEDER Funds through "Programa Operacional Factores de Competitividade-COMPETE'' and by Portuguese Funds through FCT- "Funda ao para a Ciencia e a Tecnologia", within the Project Est-C/MAT/UI0013/2011
Blow-up and finite time extinction for p(x, t)-curl systems arising in electromagnetism
"Available online 22 March 2016"We study a class of -curl systems arising in electromagnetism, with a nonlinear source term. Denoting by the magnetic field, the source term considered is of the form where : when we consider and for we have .
We introduce a suitable functional framework and a convenient basis that allow us to apply the Galerkin's method and prove existence of local or global solutions, depending on the values of and .
We study the finite time extinction or the stabilization towards zero of the solutions when and the blow-up of local solutions when .The research was partially supported by the Research Center CMAF-CIO of the University of Lisbon, Portugal, by the Research Center CMAT of the University of Minho, Portugal, with the Portuguese Funds from the "Fundacao para a Ciencia e a Tecnologia", through the Projects UID/MAT/04561/2013 and PEstOE/MAT/UI0013/2014, respectively, and by the Grant No. 15-11-20019 of the Russian Science Foundation, Russia
Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system
In this paper, we first prove the unique global strong solution with vacuum
to the two dimensional nonhomogeneous incompressible MHD system, as long as the
initial data satisfies some compatibility condition. As a corollary, the global
existence of strong solution with vacuum to the 2D nonhomogeneous
incompressible Navier-Stokes equations is also established. Our main result
improves all the previous results where the initial density need to be strictly
positive. The key idea is to use some critical Sobolev inequality of
logarithmic type, which is originally due to Brezis-Wainger.Comment: 16 page
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